A large open meadow near your city is shrinking in size because developers have started building new homes there. The relationship between $A$, the area of the meadow, in hectares, and $t$, the elapsed time, in months, since the construction began is modeled by the following function. A = 1562.5 ⋅ 10 − 0.1 t A=1562.5\cdot 10\^{-0.1t} How many months of construction will there be before the area of the meadow decreases to $500$ hectares? Give an exact answer expressed as a base- $10$ logarithm. months
Thinking about the problem We want to know how many months, $t$, it will take before the area of the meadow, $A$, decreases to $500$ hectares. So we need to find the value of $t$ for which $A=500$. Substituting $500$ in for $A$ in the model gives us the following equation. 500 = 1562.5 ⋅ 10 − 0.1 t 500=1562.5\cdot 10\^{{-0.1t}} Solving the equation We can solve the equation as shown below. 1562.5 ⋅ 10 − 0.1 t 10 − 0.1 t − 0.1 t t = 500 = 0.32 = log ( 0.32 ) = − 10 log ( 0.32 ) \begin{aligned}1562.5\cdot 10\^{{-0.1t}}&=500\\\\ 10\^{{-{0.1t}}}&=0.32\\\\ -0.1t&=\log\left(0.32\right)\\\\ t&=-10\log\left(0.32\right)\\\\ \end{aligned} It will take $-10\log\left(0.32\right)$ months for the area of the meadow to decrease to $500$ hectares. The expression above represents an exact solution to the equation. We can use a calculator to approximate the value of the expression, but this will be a rounded inexact answer. The answer The answer is $-10\log\left(0.32\right)$ months.